Regularity of monoids under Schutzenberger products

TL;DR

This paper investigates the conditions under which Schutzenberger products of monoids are regular, addressing an open problem and exploring a new variant of the product to understand its regularity properties.

Contribution

It provides partial answers to the regularity of Schutzenberger products and introduces analysis for a new version of the product.

Findings

Partial characterization of regularity conditions

Insights into the new Schutzenberger product variant

Extension of previous regularity results

Abstract

In this paper we give a partial answer to the problem which is about the regularity of Schutzenberger products in semigroups asked by Gallagher in his thesis [Problem 6.1.6]{Gallagher} and, also, we investigate the regularity for the new version of the Schutzenberger product which was defined in [Ates].